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Q.

Which of the following equations has two distinct real roots?


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a

2 x 2 −3 2 x+ 9 4 =0  

b

x 2 +x−5=0  

c

x 2 +3x+2 2 =0  

d

5 x 2 −3x+1=0      

answer is B.

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Detailed Solution

Given equations are 2 x 2 −3 2 x+ 9 4 =0,   x 2 +x−5=0,   x 2 +3x+2 2 =0,   and 5 x 2 −3x+1=0.  
For a quadratic equation a x 2 +bx+c=0  , a≠0  , there exist two distinct real roots if the discriminant, D= b 2 −4ac>0  .
On comparing 2 x 2 −3 2 x+ 9 4 =0  with a x 2 +bx+c=0  , we get a=2,b=−3 2 ,c= 9 4  .
So, the discriminant will be,
D= b 2 −4ac = (3 2 ) 2 −4(2) 9 4 =18−18 =0   The equation 2 x 2 −3 2 x+ 9 4 =0   has equal and real roots.
On comparing x 2 +x−5=0   with a x 2 +bx+c=0  , we get, a= 1, b= 1, c= -5.
So, the discriminant will be,
D= b 2 −4ac = (1) 2 −4(1) −5 =1+20 =21>0  
The equation x 2 +x−5=0   has two distinct real roots.
On comparing x 2 +3x+2 2 =0   with a x 2 +bx+c=0  , we get a=1,b=3,c=2 2  .
So, the discriminant will be,
D= b 2 −4ac = (3) 2 −4(1) 2 2 =9−8 2 <0   The equation x 2 +3x+2 2 =0   has no real roots.
On comparing 5 x 2 −3x+1=0   with a x 2 +bx+c=0  , we get, a= 5, b= -3, c=1.
So, the discriminant will be,
D= b 2 −4ac = (3) 2 −4(1) 5 =9−20 =−11<0   The equation 5 x 2 −3x+1=0   has no real roots.
Thus, the only equation with real and distinct roots is x 2 +x−5=0  .
Hence, the correct option is 2.
 
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